The D-Orbifold Programme, with Applications to Moduli Spaces of J-Holomorphic Curves Lecture 1 of 5: Overview

The d-orbifold programme, with applications to moduli spaces of J-holomorphic curves Lecture 1 of 5: Overview

Dominic Joyce, Oxford University

May 2014 Mainly work in progress. Some partial references: For d-manifolds and d-orbifolds, see arXiv:1206.4207 (survey), arXiv:1208.4948, and preliminary version of book available at http://people.maths.ox.ac.uk/∼joyce/dmanifolds.html. For C ∞ geometry, see arXiv:1104.4951 (survey), and arXiv:1001.0023.

1 / 30 Dominic Joyce, Oxford University Lecture 1: Overview

Plan of talk:

1 Introduction

2 D-manifolds and d-orbifolds

3 D-orbifolds as representable 2-functors, and moduli spaces

4 ‘Stratiﬁed manifolds’, including singular curves in moduli spaces

5 D-orbifold (co)homology, and virtual classes/cycles/chains

2 / 30 Dominic Joyce, Oxford University Lecture 1: Overview Introduction D-manifolds and d-orbifolds D-orbifolds as representable 2-functors, and moduli spaces ‘Stratiﬁed manifolds’, including singular curves in moduli spaces D-orbifold (co)homology, and virtual classes/cycles/chains 1. Introduction

These lectures concern new classes of geometric objects I call d-manifolds and d-orbifolds — ‘derived’ smooth manifolds, in the sense of Derived Algebraic Geometry. Some properties: D-manifolds form a strict 2-category dMan. That is, we have objects X, the d-manifolds, 1-morphisms f, g : X → Y, the smooth maps, and also 2-morphisms η : f ⇒ g. Smooth manifolds embed into d-manifolds as a full (2)-subcategory. So, d-manifolds generalize manifolds. There are also 2-categories dManb, dManc of d-manifolds with boundary and with corners, and orbifold versions dOrb, dOrbb, dOrbc of these, d-orbifolds. Much of diﬀerential geometry extends nicely to d-manifolds: submersions, immersions, orientations, submanifolds, transverse ﬁbre products, cotangent bundles, . . . .

3 / 30 Dominic Joyce, Oxford University Lecture 1: Overview

Introduction D-manifolds and d-orbifolds D-orbifolds as representable 2-functors, and moduli spaces ‘Stratified manifolds’, including singular curves in moduli spaces D-orbifold (co)homology, and virtual classes/cycles/chains Almost any moduli space used in any enumerative invariant problem over R or C has a d-manifold or d-orbifold structure, natural up to equivalence. There are truncation functors to d-manifolds and d-orbifolds from structures currently used — Kuranishi spaces, polyfolds, C-schemes or Deligne–Mumford C-stacks with obstruction theories. Combining these truncation functors with known results gives d-manifold/d-orbifold structures on many moduli spaces. I will also outline an approach to prove existence of d-manifold/d-orbifold structures on moduli spaces directly, using ‘representable 2-functors’. Virtual classes/cycles/chains can be constructed for compact oriented d-manifolds and d-orbifolds. So, d-manifolds and d-orbifolds provide a unified framework for studying enumerative invariants and moduli spaces. They also have other applications, and are interesting for their own sake. 4 / 30 Dominic Joyce, Oxford University Lecture 1: Overview Introduction D-manifolds and d-orbifolds D-orbifolds as representable 2-functors, and moduli spaces ‘Stratified manifolds’, including singular curves in moduli spaces D-orbifold (co)homology, and virtual classes/cycles/chains In the rest of this talk I want to discuss four different ‘big ideas’, which are the subjects of Lectures 2–5: 2 D-manifolds and d-orbifolds: what are they, why are they a 2-category, how are they related to other classes of spaces (e.g. Kuranishi spaces and polyfolds)? 3 A new approach to moduli problems using ‘representable 2-functors’ to define a d-orbifold structure on a moduli space. 4 ‘Stratified manifolds’, a new analytic tool for dealing with ‘bubbling’, ‘neck stretching’ and ‘nodes’ for J-holomorphic curves, and compactification of moduli spaces in general. 5 ‘D-orbifold (co)homology’, new (co)homology theories ∗ dH∗(Y , R), dH (Y , R) of a manifold or orbifold Y , isomorphic to ordinary (co)homology, but in which the (co)chains are 1-morphisms f : X → Y for X a compact, oriented d-orbifold with corners, plus extra data. Forming virtual classes for moduli spaces in d-orbifold (co)homology is almost trivial.

5 / 30 Dominic Joyce, Oxford University Lecture 1: Overview

In very general terms, we want to solve the following: Problem Find a (hopefully canonical) geometric structure G on moduli spaces of J-holomorphic curves M, such that (compact, oriented) spaces M with structure G can be ‘counted’ in Z, Q or some (co)homology theory, so that we can do Gromov–Witten invariants, Lagrangian Floer cohomology, Symplectic Field Theory, . . .

I claim that the structure G must be ‘derived’ in the sense of Derived Algebraic Geometry of Lurie, To¨en–Vezzosi, . . . , and that a basic understanding of derived geometry really helps here.

6 / 30 Dominic Joyce, Oxford University Lecture 1: Overview Introduction D-manifolds and d-orbifolds D-orbifolds as representable 2-functors, and moduli spaces ‘Stratiﬁed manifolds’, including singular curves in moduli spaces D-orbifold (co)homology, and virtual classes/cycles/chains Derived schemes, deformations, and obstructions Consider a moduli space M of some objects Σ (e.g. J-holomorphic curves). Linearizing the deformation theory at Σ ∈ M, we have vector spaces of deformations DΣ (typically the kernel of an operator LΣ) and obstructions OΣ (typically the cokernel of LΣ). If we model M as a classical space (e.g. a scheme or stack), then M remembers DΣ as the tangent space TΣM, but forgets OΣ. If we model M as a derived space (e.g. a derived scheme or 0 stack), then M remembers both DΣ, OΣ, as DΣ = H (TM|Σ), 1 OΣ = H (TM|Σ) for TM the tangent complex of M. Derived geometry was introduced exactly to give geometric structures which remember all the deformation theory. To ‘count’ moduli spaces M of J-holomorphic curves correctly, the geometric structure G on M must encode the obstructions OΣ as well as the deformations DΣ. So G must be ‘derived’.

7 / 30 Dominic Joyce, Oxford University Lecture 1: Overview

Suppose we have a category C (e.g. a category of complexes), and we want to invert (localize) some class of morphisms W in C called ‘weak equivalences’ (e.g. quasi-isomorphisms) which are not isomorphisms, to form a new ‘derived’ category C¯. A fundamental insight in derived geometry is that C¯ should be a higher category (usually an ∞-category), not an ordinary category. Derived objects (e.g. derived schemes) always form higher categories. You can truncate to an ordinary category, but you lose too much information (e.g. the universal property for ﬁbre products of derived objects only makes sense in the higher category). D-manifolds and d-orbifolds form 2-categories, the simplest kind of higher category.

8 / 30 Dominic Joyce, Oxford University Lecture 1: Overview Introduction D-manifolds and d-orbifolds D-orbifolds as representable 2-functors, and moduli spaces ‘Stratiﬁed manifolds’, including singular curves in moduli spaces D-orbifold (co)homology, and virtual classes/cycles/chains 2.2. Other classes of spaces. Kuranishi spaces versus d-orbifolds

A Kuranishi structure on a space M (Fukaya–Oh–Ohta–Ono) involves Kuranishi neighbourhoods (Vp, Ep, sp) and (non-invertible) coordinate changes (fpq, fˆpq):(Vp, Ep, sp) → (Vq, Eq, sq). Think of the Kuranishi neighbourhoods (Vp, Ep, sp) as the objects in a category C, and the coordinate changes (fpq, fˆpq) as the weak equivalences W we want to invert. Derived geometry says that to do the job properly, we must use higher categories, but FOOO do not. This is the source of some of the problems in the theory. I regard d-orbifolds as the ‘correct’ deﬁnition of Kuranishi spaces, what Kuranishi spaces should have been. It should be possible to give a FOOO-style deﬁnition of Kuranishi spaces involving 2-categories, and get a 2-category equivalent to d-orbifolds dOrb.

9 / 30 Dominic Joyce, Oxford University Lecture 1: Overview

Polyfolds (Hofer–Wysocki–Zehnder) do not use higher categories, so far as I know. They get away with this because they never localize. Therefore a polyfold comprises a huge amount of information, the whole of the functional-analytic moduli problem, as nothing is forgotten by localizing. This works, but I feel polyfolds are unwieldy, and unsatisfying as geometric spaces. • Schemes M with obstruction theory φ : E → LM in algebraic geometry (Behrend–Fantechi) work very nicely — much more so than Kuranishi spaces or polyfolds in the (rather more diﬃcult) diﬀerential-geometric context. Note that they are ‘derived’ objects (as I said M must be), since E• is an object in the derived category Db coh(M), and should be understood as the cotangent complex LM of an underlying derived scheme M.

10 / 30 Dominic Joyce, Oxford University Lecture 1: Overview Introduction D-manifolds and d-orbifolds D-orbifolds as representable 2-functors, and moduli spaces ‘Stratiﬁed manifolds’, including singular curves in moduli spaces D-orbifold (co)homology, and virtual classes/cycles/chains 2.3. D-manifolds – sketch of the deﬁnition

Algebraic geometry (based on algebra and polynomials) has excellent tools for studying singular spaces – the theory of schemes. In contrast, conventional differential geometry (based on smooth real functions and calculus) deals well with nonsingular spaces – manifolds – but poorly with singular spaces. There is a little-known theory of schemes in differential geometry, C ∞-schemes, going back to Lawvere, Dubuc, Moerdijk and Reyes, . . . in synthetic differential geometry in the 1960s-1980s. C ∞-schemes are essentially algebraic objects, on which smooth real functions and calculus make sense. Our d-manifolds are a special kind of derived C ∞-scheme, combining C ∞-algebraic geometry and derived geometry.

11 / 30 Dominic Joyce, Oxford University Lecture 1: Overview

Let X be a manifold, and write C ∞(X ) for the smooth functions ∞ c : X →R. Then C (X ) is an R-algebra: we can add smooth functions (c, d) 7→ c + d, and multiply them (c, d) 7→ cd, and multiply by λ ∈ R. But there are many more operations on C ∞(X ) than this, e.g. if c : X → R is smooth then exp(c): X → R is smooth, giving exp : C ∞(X ) → C ∞(X ), which is algebraically independent of addition and multiplication. n ∞ n ∞ Let f : R → R be smooth. Deﬁne Φf : C (X ) → C (X ) by Φf (c1,..., cn)(x) = f c1(x),..., cn(x) for all x ∈ X . Then 2 addition comes from f : R → R, f :(x, y) 7→ x + y, multiplication from (x, y) 7→ xy, etc. This huge collection of algebraic operations ∞ ∞ Φf make C (X ) into an algebraic object called a C -ring.

12 / 30 Dominic Joyce, Oxford University Lecture 1: Overview Introduction D-manifolds and d-orbifolds D-orbifolds as representable 2-functors, and moduli spaces ‘Stratiﬁed manifolds’, including singular curves in moduli spaces D-orbifold (co)homology, and virtual classes/cycles/chains

Deﬁnition ∞ n A C -ring is a set C together with n-fold operations Φf : C → C n for all smooth maps f : R → R, n > 0, satisfying: n m Let m, n > 0, and fi : R → R for i = 1,..., m and g : R → R n be smooth functions. Deﬁne h : R → R by h(x1,..., xn) = g(f1(x1,..., xn),..., fm(x1 ..., xn)), n for (x1,..., xn) ∈ R . Then for all c1,..., cn in C we have

Φh(c1,..., cn) = Φg (Φf1 (c1,..., cn),..., Φfm (c1,..., cn)). Also deﬁning πj :(x1,..., xn) 7→ xj for j = 1,..., n we have

Φπj :(c1,..., cn) 7→ cj . A morphism of C ∞-rings is φ : C → D with n n n Φf ◦ φ = φ ◦ Φf : C → D for all smooth f : R → R. Write C∞Rings for the category of C ∞-rings.

13 / 30 Dominic Joyce, Oxford University Lecture 1: Overview

We can now develop the whole machinery of scheme theory and stack theory in algebraic geometry, replacing rings or algebras by C ∞-rings throughout — see my arXiv:1104.4951, arXiv:1001.0023. This gives a category of C ∞-schemes C∞Sch and a 2-category of C ∞-stacks C∞Sta, which contain manifolds Man ⊂ C∞Sch and orbifolds Orb ⊂ C∞Sta as full (2-)subcategories. We also deﬁne (quasi)coherent sheaves on C ∞-schemes and C ∞-stacks, generalizing vector bundles on manifolds and orbifolds.

14 / 30 Dominic Joyce, Oxford University Lecture 1: Overview Introduction D-manifolds and d-orbifolds D-orbifolds as representable 2-functors, and moduli spaces ‘Stratiﬁed manifolds’, including singular curves in moduli spaces D-orbifold (co)homology, and virtual classes/cycles/chains Derived C ∞-algebraic geometry

Derived smooth manifolds (or orbifolds) should be defined as special examples of suitable higher categories of derived C ∞-schemes or derived C ∞-stacks. This was the approach taken by David Spivak (arXiv:0810.5175, Duke Math. J.), a student of Jacob Lurie, who defined an ∞-category of simplicial-C ∞-ringed spaces, with an ∞-subcategory of ‘derived manifolds’. My approach is roughly a simplified 2-category truncation of Spivak’s (for the precise relation, see Borisov arXiv:1212.1153). I define 2-categories of d-spaces dSpa and d-stacks dSta containing d-manifolds dMan ⊂ dSpa and d-orbifolds dOrb ⊂ dSta as full 2-subcategories.

15 / 30 Dominic Joyce, Oxford University Lecture 1: Overview

A derived scheme (dg-scheme) X is roughly a topological space X equipped with a sheaf (or homotopy sheaf) OX of dg-rings, with points of X corresponding to prime ideals of the dg-rings. Similarly, a derived C ∞-scheme X should roughly be a topological ∞ space X equipped with a (homotopy?) sheaf OX of dg C -rings (or possibly simplicial C ∞-rings), with points of X corresponding to ideals of the dg-rings with residue ﬁeld R. A d-space X is a topological space X with a sheaf (not a homotopy ∞ • sheaf) OX of dg-C -rings C of a special kind: they are two-step dg-C ∞-rings C−1 −→d C0 such that C−1 · d[C−1] = 0, which implies that d[C−1] is a square zero ideal in the (ordinary) C ∞-ring C0, and C−1 is a module over the ‘classical’ C ∞-ring H0(C•) = C0/d[C−1].

16 / 30 Dominic Joyce, Oxford University Lecture 1: Overview Introduction D-manifolds and d-orbifolds D-orbifolds as representable 2-functors, and moduli spaces ‘Stratiﬁed manifolds’, including singular curves in moduli spaces D-orbifold (co)homology, and virtual classes/cycles/chains What is a d-manifold?

We have inclusions of (2-)categories Man ⊂ C∞Sch ⊂ dSpa, so manifolds are examples of d-spaces. A d-manifold X of virtual dimension n ∈ Z is a d-space X whose topological space X is Hausdorff and second countable, and such that X is covered by open d-subspaces Y ⊂ X with equivalences Y ' U ×g,W ,h V , where U, V , W are manifolds with dim U + dim V − dim W = n, and g : U → W , h : V → W are smooth maps, and U ×g,W ,h V is the fibre product in the 2-category dSpa. (The 2-category structure is essential to define the fibre product here.) Alternatively, we can write the local models as Y ' V ×0,E,s V , where V is a manifold, E → V a vector bundle, s : V → E a smooth section, and n = dim V − rank E. Then (V , E, s) is a Kuranishi neighbourhood on X (compare with Kuranishi spaces). We call such V ×0,E,s V affine d-manifolds.

17 / 30 Dominic Joyce, Oxford University Lecture 1: Overview

Recall the Grothendieck approach to moduli spaces in algebraic

geometry, using moduli functors. Write SchC for the category of -schemes, and Schaff for the subcategory of affine -schemes. C C C Any -scheme X defines a functor Hom(−, X ): Schop → Sets C C mapping each C-scheme S to the set Hom(S, X ). By the Yoneda Lemma, the C-scheme X is determined up to isomorphism by the functor Hom(−, X ) up to natural isomorphism. This still holds if we restrict to Schaff . Thus, given a functor F :(Schaff )op → Sets, C C we can ask if there exists a C-scheme X (necessarily unique up to canonical isomorphism) with F =∼ Hom(−, X ). If so, we call F a representable functor. More generally, Artin C-stacks are defined as a class of functors F :(Schaff )op → Groupoids. C

18 / 30 Dominic Joyce, Oxford University Lecture 1: Overview Introduction D-manifolds and d-orbifolds D-orbifolds as representable 2-functors, and moduli spaces ‘Stratiﬁed manifolds’, including singular curves in moduli spaces D-orbifold (co)homology, and virtual classes/cycles/chains Grothendieck’s moduli schemes

Suppose we have an algebro-geometric moduli problem (e.g. vector bundles on a smooth projective C-scheme Y ) for which we want to form a moduli scheme. Grothendieck tells us that we should define a moduli functor F :(Schaff )op → Sets, such that C for each affine C-scheme S, F (S) is the set of isomorphism classes of families of the relevant objects over S (e.g. vector bundles over Y × S). Then we should try to prove F is a representable functor, using some criteria for representability. If it is, F =∼ Hom(−, M), where M is the (fine) moduli scheme. To form a moduli stack, we define F :(Schaff )op → Groupoids, so C that for each affine C-scheme S, F (S) is the category of families of objects over S, with morphisms isomorphisms of families, and try to show F satisfies the criteria to be an Artin stack.

19 / 30 Dominic Joyce, Oxford University Lecture 1: Overview

D-orbifolds dOrb are a 2-category with all 2-morphisms invertible. Thus, if S, X ∈ dOrb then Hom(S, X) is a groupoid, and Hom(−, X): dOrbop → Groupoids is a 2-functor, which determines X up to equivalence in dOrb. This is still true if we restrict to affine d-manifolds dManaff ⊂ dOrb. Thus, we can consider 2-functors F :(dManaff )op → Groupoids, and ask whether there exists a d-orbifold X (unique up to equivalence) with F ' Hom(−, X). If so, we call F a representable 2-functor. I expect there are nice criteria for when F is representable: (A) F satisfies a sheaf-type condition; (B) the ‘coarse topological space’ M = F (point)/isos is Hausdorff and second countable; (C) F admits a ‘Kuranishi neighbourhood’ of dimension n ∈ Z near each x ∈ M, a local model with a universal property.

20 / 30 Dominic Joyce, Oxford University Lecture 1: Overview Introduction D-manifolds and d-orbifolds D-orbifolds as representable 2-functors, and moduli spaces ‘Stratiﬁed manifolds’, including singular curves in moduli spaces D-orbifold (co)homology, and virtual classes/cycles/chains Moduli 2-functors in diﬀerential geometry

Suppose we are given a moduli problem in differential geometry (e.g. J-holomorphic curves in a symplectic manifold) and we want to form a moduli space M as a d-orbifold. I propose that we should define a moduli 2-functor F :(dManaff )op → Groupoids, such that for each affine d-manifold S, F (S) is the category of families of the relevant objects over S. Then we should try to prove F satisfies (A)–(C), and so is represented by a d-orbifold M; here (A),(B) will usually be easy, and (C) the difficult part. To define F , we need a good notion of ‘families of J-holomorphic curves over a base S, an affine d-manifold’. I will explain a short, natural definition for F for J-holomorphic curves in Lecture 3.

21 / 30 Dominic Joyce, Oxford University Lecture 1: Overview

Some remarks: Current definitions of differential-geometric moduli spaces (e.g. Kuranishi spaces, polyfolds) are generally long, complicated ad hoc constructions, with no obvious naturality. In contrast, if we allow differential geometry over d-manifolds, my approach gives you a short, natural definition of the moduli functor F , followed by a long proof that F is representable. The effort moves from a construction to a theorem. The definition of F involves only finite-dimensional families of smooth objects, with no analysis, Banach spaces, etc. (But the proof of (C) will involve analysis and Banach spaces.) This enables us to sidestep some analytic problems, e.g. non-smoothness of action of diffeomorphisms on Banach spaces of maps (cf. sc-smoothness in polyfold theory).

22 / 30 Dominic Joyce, Oxford University Lecture 1: Overview Introduction D-manifolds and d-orbifolds D-orbifolds as representable 2-functors, and moduli spaces ‘Stratiﬁed manifolds’, including singular curves in moduli spaces D-orbifold (co)homology, and virtual classes/cycles/chains 4. ‘Stratiﬁed manifolds’, and curves with nodes

In symplectic geometry, one considers moduli spaces M of J-holomorphic curves u :Σ → S, for (S, ω) a symplectic manifold with almost complex structure J. For counting problems (Gromov–Witten, etc.) it is essential that M be compact. But if we consider only nonsingular Riemann surfaces Σ, we generally get noncompact M. To get compactified moduli spaces M, we must include Riemann surfaces Σ with singularities (nodes). Two closely related problems are moduli spaces of J-holomorphic curves with ends in Symplectic Field Theory including curves which ‘stretch’ along an infinite cylinder, and (simpler) moduli spaces of gradient flow lines in Morse homology, including ‘broken flow lines’.

23 / 30 Dominic Joyce, Oxford University Lecture 1: Overview

Modelling such moduli spaces M analytically near Σ ∈ M with nodes is rather messy, and apparently not smooth: the obvious constructions yield Kuranishi neighbourhoods (V , E, s) on M in which the section s : V → E is not smooth normal to the nodal stratum Vnode ⊂ V , but only continuous. (Curiously, the algebraic geometry version does not suﬀer from this problem.) Non-smooth sections s : V → E would be bad in our C ∞-geometry approach. In the polyfold picture, one deals with this using a gluing proﬁle ϕ: basically, one changes the smooth structure on V along Vnode in the normal directions, to get a new manifold V˜ with the same topological space as V , such that s is smooth w.r.t. the smooth ˜ ˜ structure on V . Roughly, V = V ×r 2,[0,∞),ϕ [0, ∞), where r : V → [0, ∞) is the distance from Vnode, and ϕ : [0, ∞) → [0, ∞) is ϕ(0) = 0, ϕ(x) = e−1/x , x > 0.

24 / 30 Dominic Joyce, Oxford University Lecture 1: Overview Introduction D-manifolds and d-orbifolds D-orbifolds as representable 2-functors, and moduli spaces ‘Stratiﬁed manifolds’, including singular curves in moduli spaces D-orbifold (co)homology, and virtual classes/cycles/chains

In the ‘representable 2-functor’ approach of §3, I expect this problem comes up as follows: one can define a moduli functor F : dManaff → Groupoids for M, including curves with nodes, in a straightforward way. But F may not be representable near singular curves Σ in M. To deal with this, one should modify F using a gluing profile ϕ to get a new, representable functor F˜. However, I wish to propose an alternative approach I believe is analytically more natural; my ideas here are still at an early stage. I want to define a new category Manst of ‘stratified manifolds’, which are roughly manifolds V with designated ‘strata’ Vnode ⊂ V (e.g. the boundary of V ) such that the smooth structure on V along Vnode in the normal directions is exotic, nonstandard; so V is not actually a conventional smooth manifold near Vnode.

25 / 30 Dominic Joyce, Oxford University Lecture 1: Overview

These stratified manifolds should be included in d-manifolds and d-orbifolds to give 2-categories dManst, dOrbst. A choice of gluing profile ϕ should induce ‘smoothing functors’ from Manst, dManst, dOrbst to Manc, dManc, dOrbc. I hope that this class of analytic moduli problems can be more naturally modelled using stratified manifolds, yielding Kuranishi neighbourhoods (V , E, s) for M in which V is a stratified manifold, with strata at the points representing nodal curves, and canonical moduli functors which are representable in dOrbst, though not necessarily in dOrbc. There appear to be connections with the work of Richard Melrose on analysis on manifolds with boundary and corners. For more details, see Lecture 4.

26 / 30 Dominic Joyce, Oxford University Lecture 1: Overview Introduction D-manifolds and d-orbifolds D-orbifolds as representable 2-functors, and moduli spaces ‘Stratiﬁed manifolds’, including singular curves in moduli spaces D-orbifold (co)homology, and virtual classes/cycles/chains 5. D-orbifold (co)homology, and virtual cycles

Note: you can ﬁnd a previous version of this project using Kuranishi spaces instead of d-orbifolds at arXiv:0710.5634 (survey), and arXiv:0707.3572. These papers should not be trusted, as the Kuranishi spaces material is dodgy, but they do show the basic ideas. Once we have given moduli spaces of J-holomorphic curves M the structure of d-orbifolds (or whatever), to do Gromov–Witten invariants, Lagrangian Floer cohomology, Symplectic Field Theory, . . . , we need to associate a virtual class (or virtual cycle, or virtual chain) to M, in some (co)homology theory. That is, we need a bridge between moduli spaces and homological algebra.

27 / 30 Dominic Joyce, Oxford University Lecture 1: Overview

In [FOOO], this is done by perturbing the Kuranishi spaces using multisections to get a (Q-weighted, non-Hausdorff) manifold, triangulating this by simplices to get a chain in singular homology. This process is acutely painful, because singular homology does not play at all nicely with Kuranishi spaces, and much of the algebraic complexity of [FOOO] is due to the problems this causes – especially, perturbing Kuranishi spaces to transverse. I propose to define new (co)homology theories dH∗(Y , R), dH∗(Y , R) of a manifold or orbifold Y , isomorphic to ordinary (co)homology, but in which the (co)chains are 1-morphisms f : X → Y for X a compact, oriented d-orbifold with corners, plus extra data. Forming virtual classes for moduli spaces in d-orbifold (co)homology is almost trivial, there is no need to perturb; the homological algebra in [FOOO] can be drastically simplified.

28 / 30 Dominic Joyce, Oxford University Lecture 1: Overview Introduction D-manifolds and d-orbifolds D-orbifolds as representable 2-functors, and moduli spaces ‘Stratiﬁed manifolds’, including singular curves in moduli spaces D-orbifold (co)homology, and virtual classes/cycles/chains D-orbifold homology

Let Y be a manifold or orbifold, and R a Q-algebra. We define a complex of R-modules dC∗(Y , R); ∂ , whose homology groups dH∗(Y , R) are the d-orbifold homology of Y . Chains in dCk (Y ; R) for k ∈ Z are R-linear combinations of equivalence classes [X, f, G], where X is a compact, oriented d-orbifold with corners of dimension k, f : X → Y is a 1-morphism in dManc, and G is some extra ‘gauge-fixing data’ associated to X, for which there will be many possible choices. If we did not include G then chains (X, f) might have infinite automorphism groups, leading to bad behaviour. The boundary operator ∂ : dCk (Y ; R) → dCk−1(Y ; R) maps ∂ : X, f, G 7−→ ∂X, f ◦ iX, G|∂X . Note that ∂2X has an orientation-reversing involution σ : ∂2X → ∂2X. Using this we show that ∂2 = 0.

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sing Singular homology H∗ (Y ; R) may be deﬁned using sing sing C∗ (Y ; R); ∂ , where Ck (Y ; R) is spanned by smooth maps f : ∆k → Y , for ∆k the standard k-simplex, thought of as a manifold with corners. dH sing We deﬁne an R-linear map Fsing : Ck (Y ; R) → dCk (Y ; R) by dH Fsing : f 7−→ ∆k , f , G∆k ,

with G∆k some standard choice of gauge-ﬁxing data for ∆k . dH dH dH Then Fsing ◦ ∂ = ∂ ◦ Fsing, so that Fsing induces morphisms dH sing Fsing : Hk (Y ; R) → dHk (Y ; R). I hope to show these are isomorphisms. Proving this will involve perturbing d-orbifolds to orbifolds or manifolds and triangulating by simplices — the messy bits of [FOOO]. But we only have to do this once, not every time we use the theory.

30 / 30 Dominic Joyce, Oxford University Lecture 1: Overview